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Predictions and Primitive Ontology

in Quantum Foundations:

A Study of Examples

Valia Allori∗, Sheldon Goldstein†,Roderich Tumulka‡, and Nino Zanghı§

August 21, 2012

Abstract

A major disagreement between different views about the foundations of quan-tum mechanics concerns whether for a theory to be intelligible as a fundamentalphysical theory it must involve a “primitive ontology” (PO), i.e., variables de-scribing the distribution of matter in 4-dimensional space-time. In this paper,we illustrate the value of having a PO. We do so by focussing on the role thatthe PO plays for extracting predictions from a given theory and discuss valid andinvalid derivations of predictions. To this end, we investigate a number of exam-ples based on toy models built from the elements of familiar interpretations ofquantum theory.

PACS: 03.65.Ta. Key words: quantum theory without observers; Ghirardi–Rimini–Weber (GRW) theory of spontaneous wave function collapse; Bohmian mechanics;many-worlds view of quantum mechanics.

Dedicated to Tim Maudlin on the occasion of his 50th birthday

∗Department of Philosophy, Northern Illinois University, Zulauf Hall 920, DeKalb, IL 60115, USA.E-mail: vallori@niu.edu

†Departments of Mathematics, Physics and Philosophy, Rutgers University, Hill Center, 110 Frel-inghuysen Road, Piscataway, NJ 08854-8019, USA. E-mail: oldstein@math.rutgers.edu

‡Department of Mathematics, Rutgers University, Hill Center, 110 Frelinghuysen Road, Piscataway,NJ 08854-8019, USA. E-mail: tumulka@math.rutgers.edu

§Dipartimento di Fisica dell’Universita di Genova and INFN sezione di Genova, Via Dodecaneso 33,16146 Genova, Italy. E-mail: zanghi@ge.infn.it

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Contents

1 Introduction 2

2 The GRWm and GRWf Theories 32.1 The GRW Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 GRWm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 GRWf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Predictions and Primitive Ontology 63.1 Calibration Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Taking the PO Seriously . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.3 Examples From the Literature . . . . . . . . . . . . . . . . . . . . . . . . 93.4 The Main Theorem About Operators in The GRW Formalism . . . . . . 113.5 The GRW Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 A Set of Examples 144.1 Bohmian mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.2 Bohmian Trajectories and GRW Collapses . . . . . . . . . . . . . . . . . 15

4.2.1 Bohm’s Law and GRW’s Law . . . . . . . . . . . . . . . . . . . . 154.2.2 Bohm’s Law and a Modified GRW Law . . . . . . . . . . . . . . . 164.2.3 Trajectories From the GRW Wave Function . . . . . . . . . . . . 174.2.4 Configuration Jumps and GRW Law . . . . . . . . . . . . . . . . 184.2.5 Another Way of Configuration Jumps and GRW Law . . . . . . . 18

4.3 MBM: Bohm-Like Trajectories From the Master Equation . . . . . . . . 194.3.1 Empirical Equivalence of MBM with GRWm and GRWf . . . . . 20

4.4 Master Equation and Matter Density . . . . . . . . . . . . . . . . . . . . 214.5 Master Equation and Flashes . . . . . . . . . . . . . . . . . . . . . . . . 22

5 Conclusions 24

A Proof of Equivariance in MBM 24

1 Introduction

This paper is based on the view that any viable interpretation of quantum mechanics, ormore generally any fundamental physical theory, must involve variables describing thedistribution of matter in space and time. Such variables describe the primitive ontology(PO) of the theory [3]. Examples of theories with a PO, examples very relevant to thispaper, include Bohmian mechanics [21, 12, 20] (with the PO given by the particles),Schrodinger’s first quantum theory [42, 4] (with the PO given by the mass or chargedensity), as well as GRWm and GRWf, two versions of the Ghirardi–Rimini–Weber(GRW) theory [32, 15] of spontaneous wave function collapse corresponding to twochoices of PO: the matter density ontology (GRWm) and the flash ontology (GRWf). We

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recall the definitions of GRWm and GRWf in Section 2 and that of Bohmian mechanicsin Section 4.1.

This paper concerns derivations of empirical predictions, i.e., of predictions that canbe tested empirically. We believe that, for a theory with a PO, a derivation of empiricalpredictions should be based on the PO. To elucidate this statement is the goal of thispaper. We describe how a valid derivation of empirical predictions should work andwhere it should use the PO and the laws governing it. We do so mainly by means ofexamples. Some of the examples (Section 3) are valid derivations found in the literatureconcerning GRW or other serious theories, but most of them (Section 4) are novel andconcern toy theories that we have concocted for the purposes of this paper. Some ofthese toy theories make completely wrong predictions; but that does not preclude themfrom exemplifying valid derivations of predictions (which just happen to disagree withempirical findings). For a more systematic analysis of why a fundamental physicaltheory needs to explicitly involve a PO (describing the distribution of matter in spaceand time), see [3, 38].

In the following we simply say “predictions” for “empirical predictions,” and “em-pirical content” for the sum of all empirical predictions of a theory. We do not, in thispaper, compute any specific predictions for specific experiments. For simplicity, we limitour considerations to the non-relativistic quantum mechanics of N spinless particles; themodels we describe can easily be modified so as to incorporate spin.

2 The GRWm and GRWf Theories

For introductory presentations of the idea behind theories of spontaneous wave functioncollapse, such as GRW theory, see [15, 33, 41, 30, 3]. Detailed introductions to theGRWm and GRWf theories have been given recently in [3, 46, 34]. Here we give only abrief description.

2.1 The GRW Process

In both GRWm and GRWf the evolution of the wave function follows, instead of theSchrodinger equation, a stochastic jump process in Hilbert space, called the GRW pro-cess. Consider a quantum system of (what would normally be called) N “particles,”described by a wave function ψ = ψ(q1, . . . , qN), qi ∈ R

3, i = 1, . . . , N . The GRWprocess behaves as if an “observer” outside the universe made unsharp “quantum mea-surements” of the position observable of a randomly selected particle at random timesT1, T2, . . . that occur with constant rate Nλ, where λ is a new constant of nature oforder of 10−16 s−1, called the collapse rate per particle. The wave function “collapses”at every time T = Tk, i.e., it changes discontinuously and randomly as follows. Thepost-collapse wave function ψT+ = limtցT ψt is obtained from the pre-collapse wavefunction ψT− = limtրT ψt by multiplication by a Gaussian function,

ψT+(q1, . . . , qN) =1

Zg(qI −X)1/2 ψT−(q1, . . . , qN ) , (1)

3

where

g(x) =1

(2πσ2)3/2e−

x2

2σ2 (2)

is the 3-dimensional Gaussian function of width σ, I is chosen randomly from 1, . . . , N ,and

Z = Z(X) =

(∫

R3N

dq1 · · · dqN g(qI −X) |ψT−(q1, . . . , qN )|2)1/2

(3)

is a normalization factor. The width σ is another new constant of nature of order of10−7m, while the center X = Xk is chosen randomly with probability density ρ(x) =Z(x)2. We will refer to (Xk, Tk) as the space-time location of the collapse.

Between the collapses, the wave function evolves according to the Schrodinger equa-tion corresponding to the standard Hamiltonian H governing the system, e.g., given, forN spinless particles, by

H = −N∑

k=1

~2

2mk∇2qk+ V, (4)

where mk, k = 1, . . . , N , are the masses of the particles, and V is the potential energyfunction of the system. Due to the stochastic evolution, the wave function ψt at time tis random.

This completes our description of the GRW law for the evolution of the wave function.According to GRW theory, the wave function ψ of the universe evolves according to thisstochastic law, starting from the initial time (say, the big bang). As a consequence[7, 34], a subsystem of the universe (comprising M < N “particles”) will have a wavefunction ϕ of its own that evolves according to the appropriate M-particle version ofthe GRW process during the time interval [t1, t2], provided that ψ(t1) = ϕ(t1) ⊗ χ(t1)and that the system is isolated from its environment during that interval.

Another remark concerns density matrices. It is a standard fact that with everyprobability distribution µ(dψ) on the unit sphere

S(H ) =ψ ∈ H : ‖ψ‖ = 1

(5)

of a Hilbert space H there is associated a density matrix

ρµ = Eµ|ψ〉〈ψ| =∫

S(H )

µ(dψ) |ψ〉〈ψ| , (6)

where E means expectation. Since the GRW process is stochastic, ψt is random, andwith its distribution µt there is associated a density matrix ρt = ρµt . It turns outthat ρt evolves according to an autonomous equation (i.e., one that depends on µt onlythrough ρt), the “master equation.” This equation is of a type known as a Lindbladequation, or quantum dynamical semigroup, and tends to evolve pure states into mixedstates. Although the mathematical details play no role in this paper, we give the masterequation for the sake of completeness:

dρtdt

= − i~[H, ρt] + λ

N∑

k=1

∫d3x g

1/2k,x ρt g

1/2k,x −Nλρt . (7)

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Here, g1/2k,x is the multiplication operator by the function g(qk − x)1/2 with g the 3-

dimensional Gaussian function as in (2).

We now turn to the primitive ontology (PO). In the subsections below we present twoversions of the GRW theory, based on two different choices of the PO, namely the matterdensity ontology (GRWm in Section 2.2) and the flash ontology (GRWf in Section 2.3).

2.2 GRWm

GRWm postulates that, at every time t, matter is continuously distributed in space withdensity function m(x, t) for every location x ∈ R

3, given by

m(x, t) =

N∑

i=1

mi

∫

R3N

dq1 · · · dqN δ3(qi − x)∣∣ψt(q1, . . . , qN)

∣∣2 (8)

=N∑

i=1

mi

∫

R3(N−1)

dq1 · · · dqi−1 dqi+1 · · · dqN∣∣ψt(q1, . . . , qi−1, x, qi+1, . . . , qN)

∣∣2 . (9)

In words, one starts with the |ψ|2–distribution in configuration space R3N , then obtainsthe marginal distribution of the i-th degree of freedom qi ∈ R

3 by integrating out allother variables qj , j 6= i, multiplies by the mass associated with qi, and sums over i.Alternatively, (8) can be rewritten as

m(x, t) = 〈ψt|M(x)|ψt〉 (10)

with M(x) =∑

imi δ3(Qi−x) the mass density operator, defined in terms of the position

operators Qiψ(q1, . . . , qN) = qi ψ(q1, . . . , qN).

2.3 GRWf

According to GRWf, the PO is given by “events” in space-time called flashes, mathe-matically described by points in space-time. What this means is that in GRWf matteris neither made of particles following world lines, nor of a continuous distribution ofmatter such as in GRWm, but rather of discrete points in space-time, in fact finitelymany points in every bounded space-time region.

In the GRWf theory, the space-time locations of the flashes can be read off from thehistory of the wave function: every flash corresponds to one of the spontaneous collapsesof the wave function, and its space-time location is just the space-time location of thatcollapse. The flashes form the set

F = (X1, T1), . . . , (Xk, Tk), . . . (11)

(with T1 < T2 < . . .). Alternatively, we may postulate that flashes can be of N differenttypes (“colors”), corresponding to the mathematical description

F = (X1, T1, I1), . . . , (Xk, Tk, Ik), . . . , (12)

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with Ik the number of the particle affected by the k-th collapse.Note that if the number N of degrees of freedom in the wave function is large, as in

the case of a macroscopic object, the number of flashes is also large (if λ = 10−15 s−1 andN = 1023, we obtain 108 flashes per second). Therefore, for a reasonable choice of theparameters of the GRWf theory, a cubic centimeter of solid matter contains more than108 flashes per second. That is to say that large numbers of flashes can form macroscopicshapes, such as tables and chairs. That is how we find an image of our world in GRWf.

We should remark that the word “particle” can be misleading. According to GRWf,there are no particles in the world, just flashes and a wave function. According toGRWm, there are no particles, just continuously distributed matter and a wave function.The word “particle” should thus not be taken literally (just like, e.g., the word “sunrise”);we use it only because it is common terminology in quantum mechanics.

3 Predictions and Primitive Ontology

In the PO view, a satisfactory theory should have a PO. In this view, the PO alsopermits the derivation of predictions. For example, if we want to derive that in acertain experiment the pointer of the apparatus will end up pointing to the value zwith a certain probability then, according to the PO view, we need to derive that theconfiguration of the PO will be such that the matter of the pointer is in a configurationcorresponding to the pointer pointing to z. In contrast, it would not be appropriate inthis view to merely show that the wave function lies (approximately) in a subspace ofHilbert space corresponding to the pointer pointing to z.

For example, a number of empirical predictions of GRW theory have been derivedin [32, 40, 36, 35, 1]. It was found that the predictions deviate from those of quantummechanics but only so slightly that no experimental test has been possible so far [1, 29].However, the logical clarity of the derivations in [32, 40, 36, 35, 1] leaves something tobe desired, as they do not refer to the PO but limit themselves to analyses of the wavefunction. While we do not dispute that the claimed predictions are indeed predictionsof GRWm and GRWf, we do see a gap in the derivation. The situation is similar tothat of a calculation that yields the correct result but is not mathematically rigorous.Here, the problem is not one of mathematical rigor but of clarity—philosophical, onto-logical, conceptual, and physical clarity. We will describe in this chapter, particularlyin Section 3.4, how to close this gap.

3.1 Calibration Functions

Given an experiment E , its outcome Z is a function of the (configuration of the) PO,

Z = ζ(PO) . (13)

That is, in GRWf Z = ζGRWf(F ), and in GRWm Z = ζGRWm(m). Similarly, the POplays a key role for the claim of empirical equivalence between two theories (for whichthere will be several examples in this paper).

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Let us elaborate a bit on what the function ζ should look like. In GRWm, it isnatural that Z should be a functional of m(·, t), the distribution of matter at the timet when the experiment is completed. What the ζ function does is essentially to read offthe outcome from the display of the apparatus. For example, if the outcome is displayedby means of the position of a needle on a scale, ζ should read off from m the position ofthe needle, and may concretely be the following: Suppose the experiment is so arrangedthat the region R ⊆ R

3, for simplicity a cuboid R = [a1, b1]× [a2, b2]× [a3, b3], containsno other matter than the needle, and suppose the scale is along the x1-axis between a1and b1. Then the mean x1 coordinate of the matter distribution at time t inside R isgiven by

〈x1〉 =∫Rdx1 dx2 dx3 x1m(x1, x2, x3, t)∫Rdx1 dx2 dx3m(x1, x2, x3, t)

, (14)

and a typical choice of calibration function would be

ζ(m) = z0 + α〈x1〉 (15)

with suitable proportionality constant α. Note that ζ is a functional of m(·, t). If werequire that Z be a discrete variable, we may replace (15) with a suitable step function,such as

ζ(m) = z0 + [α〈x1〉] , (16)

where [z] denotes the nearest integer to the real number z.In GRWf, Z must depend on the history in an entire time interval, say [t, t + ∆t]

with t the time when the experiment is completed and ∆t, say, a millisecond. In theexample of the needle pointing to the outcome, ζ needs to read off the position of theneedle from the flashes and rescale it appropriately. As a concrete example, supposeagain the region R contains no other matter than the needle, and suppose the scale isalong the x1-axis between a1 and b1. Then the mean x1 coordinate of the flashes during[t, t+∆t] in R is given by

〈x1〉 =∑

k 1t≤Tk≤t+∆t 1Xk∈R (Xk)1∑k 1t≤Tk≤t+∆t 1Xk∈R

, (17)

where 1C is 1 when the condition C is satisfied and 0 otherwise. A typical choice ofcalibration function would be given in terms of this 〈x1〉 by (15) or (16).

3.2 Taking the PO Seriously

As a consequence of the view that the PO represents matter, we are forced to take thePO seriously.

For example, the transformation behavior of the PO under a Lorentz (or Galilean)transformation is constrained by its geometrical nature, and that of the non-primitiveontology (i.e., the remaining part of the ontology) is constrained by its relation tothe PO; see section 4.2 of [3] for elaboration. If the PO consists of flashes, then theflashes have to transform like space-time points under Lorentz transformations. If the

7

PO consists of particle world lines, then they have to transform like world lines, i.e., bytransforming every space-time point on the world line. Put differently, a world line mustbe an unambiguous set of space-time points, independently of the choice of coordinates.

As another example for what it means to take the PO seriously, we look at a difficultythat arose in [6] with an over-simplified discussion of how to derive in GRWm that, ina suitable situation, a pointer is pointing to z. The coordinates x1, . . . , xN ∈ R

3 of theconfiguration space R

3N of a pointer consisting of N “particles” were decomposed intothe center of mass xcm =

∑imixi/

∑jmj ∈ R

3 and relative coordinates r1, . . . , r3N−3,and it was derived that in this decomposition the wave function, in a suitable situation, isapproximately a product ψcm(xcm)ψrel(r1, . . . , r3N−3), where the first factor, the center-of-mass wave function ψcm, is a very narrow wave packet. Therefore, it was suggested,the matter density associated with the center-of-mass, mcm(x, t) = |ψcm(x, t)|2, is verynarrow, too (thus making the position of the pointer sufficiently precisely defined). Thedifficulty with this argument is that mcm is not the right quantity to look at: It is notthe PO, not the matter, not real; it is just a mathematical quantity. What counts iswhether m as defined in (8) is concentrated in the right location, and there is no simplerelation between m and mcm. For example, the width of mcm is, in realistic examples,10−13 m whereas that of m is 10−3 m (= the width of the pointer). So this argumentdid not take the PO seriously. Rather, it still treated ψ in a more or less conventionalway as providing probabilities for configurations of particles, as that is the situation inwhich the extreme narrowness of ψcm can be relevant.

The issue of taking the PO seriously also arises in the context of studying the limita-tions on knowledge in GRWm and GRWf, a topic we plan to discuss in detail in a futurework [24] (and that we have outlined in [34]): Inhabitants of a GRWf or GRWm worldcannot measure the times and locations of the collapses, although these values are welldefined according to these theories. That is, some things that are real cannot always bemeasured with arbitrary accuracy. But if something cannot be measured, one may betempted to not take it seriously. So, if there are limitations to measuring the variablesrepresenting the PO, one may be tempted not to take the PO seriously. Needless tosay, we recommend resisting this temptation. The conclusion “unobservable, thereforeunreal” is, of course, not a good one.

Another remark concerns the fine difference between the names “matter density”and “mass density” for the m function. The fact that its definition (8) involves thequantity mi usually called “the mass of particle i” suggests the name “mass density” form, which, as we shall argue, can be misleading. We prefer the name “matter density”because it reflects what we take to be the fundamental meaning of the m function. Letus explain.

Suppose we had postulated, instead of (8), the following formula for the m function:

m(x, t) =

N∑

i=1

ei

∫

R3N

dq1 · · · dqN δ(qi − x)∣∣ψt(q1, . . . , qN)

∣∣2 . (18)

This is the same equation as (8), but with mi, the “mass of particle i,” replaced by

8

ei, the “charge of particle i.” In this case, it would evidently not be appropriate anymore to call the m function the mass density; rather “charge density” would seem moreappropriate. At the same time, we may postulate that the m function still representsthe matter density—i.e., we may postulate that matter is distributed continuously withdensity m—and regard (18) as merely a different law for how matter is distributed.1 Inthat case, the meaning of the m function would not have changed in any way—we wouldonly have changed the law governing it. That is why this meaning is better conveyedby the name “matter density.” Moreover, the matter that we postulate in GRWm andwhose density is given by the m function does not ipso facto have any such propertiesas mass or charge; it can only assume various levels of density. For example, the mfunction is not a source of an electromagnetic field.

In addition, the terms “mass density” and “charge density” may easily suggest thatboth quantities physically coexist: Is it not natural to expect that an extended objectpossesses both a mass density and a charge density (different from each other)? In fact,it seems completely coherent (though perhaps not desirable) to postulate that the POinvolves both a mass density and a charge density (different from each other). If, in con-trast, we postulate that the PO consists of only one density—the matter density—andchoose either (8) or (18) to determine it, then only one of the two functions representssomething real (more precisely, represents the PO), whereas the other is a pure mathe-matical fiction. In other words, in GRWm with (8) as the law of m, the formula on theright hand side of (18) lacks any physical significance, just as it does in GRWf.

3.3 Examples From the Literature

We now look at earlier arguments based on the connection we are discussing betweenempirical predictions and PO. These are also examples of valid derivations of empiricalpredictions that do not not suffer from the gap we complained about (in the secondparagraph of this chapter).

1. The first example is a proof of no-signaling (i.e., the impossibility of transmittingmessages faster than light between two distant observers, each acting on and ob-serving one of two entangled quantum objects) in GRWf due to Bell [15]. Theproof shows that for two non-interacting systems (here, the system can be takento be one object together with a nearby apparatus), the marginal distribution ofthe flashes pertaining to system 1 depends on the entangled wave function onlythrough its reduced density matrix (with system 2 traced out); nor does it dependon the Hamiltonian of system 2. Thus, it does not depend upon any messagethat observer 2 may wish to transmit. Now, since outcomes of experiments are

1It may be a difficulty with this postulate that (18) can be negative, while it is normal to think ofmatter density as non-negative. This difficulty might be addressed by adding a sufficiently large positiveconstant on the right hand side, or perhaps just by insisting that matter density can be negative, forexample representing some kind of “anti-matter” (not necessarily related to the usual meaning of thisword). For the purpose of this section, readers may ignore this difficulty.

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functions of the flashes, the outcome that observer 1 sees cannot depend on themessage observer 2 may have wished to transmit, qed. In Bell’s words [15],

Events in one system, considered separately, allow no inference about[. . . ] external fields at work in the other, . . . nor even about the veryexistence of the other system. There are no “messages” in one systemfrom the other. The inexplicable correlations of quantum mechanics donot give rise to signalling between noninteracting systems.

A similar proof was given in [43, 45] for a relativistic version of GRWf.

2. The second example is provided by derivations of empirical predictions from Bohmianmechanics, which can be found in many papers, e.g., [21, 13, 28, 2]. The PO ofBohmian mechanics consists of particles and their trajectories, and the outcomesof experiments are read off from the particle trajectories, and not directly fromthe wave function. The empirical content of Bohmian mechanics agrees exactlywith that of the quantum formalism “whenever the latter is unambiguous.”

3. The third example is taken from the study of superselection rules by Colin etal. [22]. For superselection rules it is crucial that some self-adjoint operators arenot observables. In the terminology of Colin et al., a “weak superselection rule”means that no experiment can distinguish between a superposition of vectors fromdifferent superselection sectors in Hilbert space and a suitable mixture thereof.This occurs only if all observables commute with all projections to superselectionsectors, because otherwise a quantum measurement of such an observable woulddistinguish between them. Colin et al. proved certain weak superselection rules forBohmian mechanics (and Bohmian versions of quantum field theory) as well as forGRWm and GRWf, and the proof uses that the outcome of any experiment is afunction of the PO (and not of the wave function). Indeed, while the PO does notpermit us to distinguish between a superposition and a mixture of contributionsfrom different sectors, the wave function would trivially permit this.

In more detail, Colin et al. first showed that (in certain situations) the distributionof the flashes in GRWf does not distinguish between a superposition and a suitablemixture (“strong superselection”), and concluded from this that the distributionof outcomes of experiments does not distinguish between them either. Due to theempirical equivalence between GRWf and GRWm, the same experiment does notdistinguish between them in GRWm. To sum up, the proof that some operatorsare not observables used that results of observations must be read off from thePO, and then proceeded with a suitable analysis of the PO.

4. Colin and Struyve discussed in [23] whether their Dirac sea model is empiricallyequivalent with orthodox quantum field theory. They suggested an unusual POcontaining a huge (or even infinite) number of particles with trajectories, one forevery electron in the Dirac sea. For the theory to make the right predictions, we

10

need that macroscopic facts such as outcomes of experiments can be read off fromthis PO—and that is what Colin and Struyve analyzed.

5. In [34] we have shown (and outlined before in [3]) that GRWm and GRWf areempirically equivalent. This is a non-trivial statement if and only if the PO istaken seriously.

6. Feldmann and Tumulka [29] considered other values for σ and λ than suggestedoriginally [32] (σ = 10−7 m, λ = 10−16 s−1) and evaluated for which points inthe σλ-plane the GRWm and GRWf theories are empirically adequate, and forwhich they are philosophically satisfactory. Their criterion for being philosophi-cally satisfactory is that the PO looks macroscopically like what humans normallyimagine reality to be like. As a consequence, for determining the philosophicallysatisfactory region they need to pay attention to the behavior of the PO.

7. In [34] we have derived a measurement formalism for GRWm and GRWf, i.e., a setof rules for computing the predictions of GRWm and GRWf for any experiment.We call it the GRW formalism; it is spelled out in Section 3.5 below. The GRWformalism is analogous to the quantum formalism, i.e., the usual set of rules formaking predictions in quantum mechanics. Both the quantum and the GRWformalism can be formulated as three rules, the first specifying how an isolatedsystem evolves, the second specifying the probability distribution of the outcomeof a given experiment, and the third specifying the change on the system’s statedepending of the outcome obtained. That is, the third rule is a “collapse rule,”concerning, however, not spontaneous collapses but the collapse induced by theintervention of a macroscopic apparatus. Put differently, the GRW formalism iswhat the quantum formalism needs to be replaced with if we pay attention to thesmall deviations from the quantum formalism predicted by GRWm and GRWf.

The derivation of the GRW formalism in [34] takes the PO seriously. We discussit to some extent in the next two subsections.

3.4 The Main Theorem About Operators in The GRW For-malism

Before talking about the full GRW formalism, we begin with a statement that is partof the GRW formalism, the main theorem about operators: With every experiment E ona system and every possible outcome z of E there is associated an operator Pz actingon the Hilbert space Hsys of the system. When E is performed on a system with wavefunction ψ, the outcome Z is random with probability distribution

P(Z = z) = 〈ψ|Pz|ψ〉 . (19)

This statement is actually true in both the quantum formalism and the GRW formalism;however, to the same experiment, the GRW formalism may assign operators PGRW

z

11

different from the PQuz assigned by the quantum formalism. Formulas for PQu

z and PGRWz

are given in [34]. (We assume that every experiment has a discrete set of outcomes z.)The operator Pz in (19) may be a projection, but in general it is merely a positive

operator (even in the quantum formalism). If the Pz are projections for all z, and ifthe z are real numbers, then the family Pz corresponds to the self-adjoint operatorA =

∑z z Pz, usually (but misleadingly) called “the observable measured” by E . The z

are the eigenvalues of A, and the Pz the projections to the eigenspaces. When the Pzare not projections, the family Pz forms a positive-operator-valued measure (POVM)(meaning, in this discrete case, that the positive operators Pz are such that

∑z Pz = I,

the identity operator), a concept well known in quantum information theory.

As an example of a valid derivation of predictions that takes the PO seriously, wenow outline the derivation of the main theorem about operators from GRWf, following[34]. The key point is that we take the outcome Z of the experiment to be a functionof the pattern F = (X1, T1, I1), . . . , (Xk, Tk, Ik), . . . of flashes,

Z = ζ(F ) . (20)

(It would even be realistic to assume that Z depends only on the flashes of the apparatusduring a short time interval appropriate for reading off the outcome, and not those ofthe system or of the rest of the world, and not those at earlier or later times; but thisrestriction is not needed here.)

It is a known fact [47] that the joint distribution of all flashes after time t, conditionalon all flashes up to t, depends quadratically on Ψt, the wave function of the universe attime t. Explicitly, it is given by

P(F>t ∈ S|F≤t) = 〈Ψt|Gt(S)|Ψt〉 (21)

with S any set of flash histories after t and Gt(·) a suitable POVM on the space of allflash histories.2

Let t be the time at which the experiment begins. Consider splitting the universeinto a system (the object of the experiment) and its environment (the rest of the world,including all relevant apparatuses of the experiment), corresponding to a splitting of theHilbert space into H = Hsys ⊗ Henv. We assume independence between the systemand the environment immediately before t, so that

Ψt = ψ ⊗ φ . (22)

Here we regard φ as fixed, while ψ, the initial state of the system upon which theexperiment is performed, is allowed to vary in the system Hilbert space Hsys. We maythink of φ as part of the characterization of the experiment, although in practice a

2On a continuous space Ω, a POVM associates by definition a positive operator G(S) with every(measurable) subset S ⊆ Ω in such a way that G(Ω) = I and G(S1 ∪ S2) = G(S1) + G(S2) whenS1 ∩ S2 = ∅ (and likewise for countable families of pairwise disjoint sets). For discrete Ω, G(S) can beexpressed as

∑z∈S

Pz with Pz = G(z).

12

repetition of the experiment will not begin with exactly the same wave function φ of theapparatus.

Therefore, the distribution of the random outcome Z is given by

P(Z = z) = P(F ∈ ζ−1(z)

)= 〈Ψt|Gt

(ζ−1(z)

)|Ψt〉 = 〈ψ|PGRW

z |ψ〉 , (23)

where the first scalar product is taken in the Hilbert space of the universe and the secondin Hsys, and P

GRWz is the POVM given by

PGRWz = 〈φ|Gf

(ζ−1(z)

)|φ〉 , (24)

where the scalar product is a partial scalar product in the Hilbert space of the environ-ment. Thus, for every experiment in GRWf, the distribution of outcomes is given by aPOVM PGRW

z , which is what we wanted to show.

3.5 The GRW Formalism

The full GRW formalism is best formulated in terms of density matrices.3 Perhapsthe most remarkable fact about the GRW formalism is that its abstract structure isidentical to that of the quantum formalism. It consists of three rules, the first sayinghow the density matrix ρt of a system evolves when the system is isolated (or, when “theobserver” is not “taking measurements”); the second (a form of the main theorem aboutoperators) saying that with every experiment (or “measurement”) there are associatedoperators, and that the probability for obtaining a particular outcome is given by atrace formula involving ρt; and the third saying how ρt should be changed after theexperiment, depending on the outcome. Here are the three rules, in both the quantumand the GRW version:

(i) A system isolated from its environment has at every time t a density matrix ρtwhich evolves in the quantum formalism according to the unitary (Schrodinger)evolution, which for a density matrix reads

dρtdt

= − i~[H, ρt] (25)

with H the system’s Hamiltonian, and in the GRW formalism according to themaster equation (7).

(ii) With every experiment E on a system and every possible outcome z of E there isassociated a positive operator Pz acting on the Hilbert space Hsys of the system.When E is performed on a system that has density matrix ρ at the beginning ofE , the outcome Z is random with probability distribution

P(Z = z) = tr(ρPz

). (26)

3That is because we allow the system under study, system 1, to be entangled with another system,system 2, which does not interact with either system 1 or the apparatus of the experiment; thus,ρ = tr2|ψ〉〈ψ|, where tr2 denotes the partial trace and ψ the joint wave function of systems 1 and2. Since ψ is usually not precisely known, it is also often convenient to take it to be random; thenρ = E tr2|ψ〉〈ψ|.

13

(iii) When the outcome Z = z, the density matrix ρ of the system gets replaced by

ρ′ =Cz(ρ)

trCz(ρ). (27)

with Cz a (completely positive) linear operation on density matrices.

In the quantum formalism for an ideal quantum measurement, the Pz are projec-tions, and

Cz(ρ) = Pz ρPz . (28)

Again, the operations C Quz provided by quantum formalism and C GRW

z by theGRW formalism may differ. Formulas for C Qu

z and C GRWz are given in [34].

4 A Set of Examples

A new and useful perspective on GRW theories arises from contrasting them with othertheories, even unreasonable ones. For this purpose we develop in this section a set ofexample theories which we obtain by combining elements of the known theories GRWm,GRWf, and Bohmian mechanics in new, sometimes playful, ways. Some of the theoriesobtained in this way make completely wrong predictions but are instructive nonethelesssince they illustrate the way in which predictions follow from a theory. Others makepredictions in agreement with known empirical facts, yet nobody would seriously proposethem as fundamental physical theories; still, they allow for illuminating comparisonswith GRWm and GRWf.

Here is a “theory construction kit.” Choose one of the three primitive ontologies ofGRWm, GRWf, and Bohmian mechanics: continuous matter density, flashes, or particleswith trajectories. Choose whether the laws governing the PO should involve a wavefunction ψ or a density matrix ρ. Then choose an evolution law for ψ or ρ, e.g., theunitary Schrodinger evolution, the stochastic GRW evolution for ψ, or the GRW masterequation (7) for ρ. Then consider simple laws for how ψ (or ρ, respectively) may governthe PO. In this way we arrive at about ten new theories.

4.1 Bohmian mechanics

To review the elements used in the theory construction kit, let us briefly recall the lawsof Bohmian mechanics. Bohmian mechanics is a (non-relativistic) theory of particles inmotion. The motion of a system of N particles is provided by their world lines t 7→ Qi(t),i = 1, . . . , N , where Qi(t) denotes the position in R

3 of the i-th particle at time t. Theseworld lines are determined by Bohm’s law of motion [21, 12, 20],

dQi

dt= vψi (Q1, . . . , QN) =

~

mi

Imψ∗∇iψ

ψ∗ψ(Q1 . . . , QN), (29)

where the wave function ψ evolves according to Schrodinger’s equation

i~∂ψ

∂t= Hψ , (30)

14

with H the usual nonrelativistic Schrodinger Hamiltonian; for spinless particles it is ofthe form (4).

An important probability distribution in Bohmian mechanics is the quantum equi-librium distribution

pψ(q) = |ψ(q)|2 . (31)

(While the distribution density is usually denoted ρ, we write p here in order to reservethe letter ρ for density matrices.) As a consequence of Bohm’s law of motion (29) andSchrodinger’s equation (30), |ψ|2 is equivariant. This means that if the configurationQ(t) = (Q1(t), . . . , QN(t)) of a system is random with distribution |ψt|2 at some timet, then this will be true also for any other time t. Because of equivariance, Bohmianmechanics reproduces the predictions of the quantum formalism for typical initial con-ditions of the universe, as discussed in detail in [27, 28].

4.2 Bohmian Trajectories and GRW Collapses

We begin with the particle ontology, with the particle trajectories governed by Bohm’slaw of motion. We consider several ways of combining this with the GRW evolution ofthe wave function or a similar one.

4.2.1 Bohm’s Law and GRW’s Law

First, suppose simply that the particles move according to the usual Bohmian law of mo-tion (29), but that ψ = ψt is the GRW wave function, so that the GRW process replacesthe unitary evolution. In this theory, which we denote GRWp1, the |ψ|2 distribution forthe configuration is not equivariant.

A world governed by this theory GRWp1 has little resemblance with our world:It behaves in a very unstable way. For example, a system with the wave functionof Schrodinger’s cat has, even before the collapse into either |dead〉 or |alive〉, a con-figuration of either a dead or a live cat—but the collapse need not agree with thatconfiguration. The cat could be alive before the collapse (i.e., its particle configurationis that of a live cat), and still the collapse could reduce ψ to a state vector close to|dead〉—with however no immediate change in the particle configuration of the cat.

At this point, the reader may feel unsure whether to conclude that just after collapsethe cat is really alive or that the cat is really dead. That is, partly, what makes thistheory philosophically useful, despite its empirical inadequacy: It nicely illustrates therole of the PO. Taking the PO seriously, we must conclude that the cat is really alive;after all, the PO in this theory consists of particles, and the particle configuration is oneof a live cat. This illustrates that the mere fact that the wave function is one of a deadcat does not, in and of itself, mean that there is a dead cat.

From that fateful collapse onwards, the configuration is guided, in the sense ofBohm’s law of motion (29), by that tiny part of ψ that remains of |alive〉 after thecollapse, and who knows what happens then. For sure, the further behavior of theconfiguration will be catastrophic. The further evolution of the configuration is not

15

simply that of a live cat, but will be disturbed by two factors: first, by the fact that theGaussian collapse factor will change the shape of |alive〉, and second, by the fact thattails of |dead〉, which reach the support of |alive〉 under the Schrodinger evolution, willdominate over the contribution from |alive〉.

What this example illustrates: First and foremost, this example illustrates how thesame wave function—the GRW wave function—can be combined with a different POthan usual, and thus helps us to get used to the distinction between the wave functionand the PO. Second, it illustrates how the particle ontology can be combined with dif-ferent laws for the wave function—the GRW law instead of the Schrodinger equationas in Bohmian mechanics. Third, the example illustrates what it means to derive pre-dictions from the PO rather than from the wave function, as the wave function is wellbehaved but the particle configuration is not. In particular, GRWp1 forces us to facethe question: Do the predictions follow from the wave function or from the PO? Fourthand finally, GRWp1 shows that the GRW wave function can be part of a theory makingcompletely different predictions than GRWm and GRWf.

Another observation: GRWp1 is presumably an example of a theory without a for-malism. Note first that the |ψ|2 distribution is not equivariant in GRWp1, in fact there isno equivariant density formula at all. This undercuts the reasons for assuming the initialconfiguration was |ψ|2 distributed, so that we lose the basis for deriving predictions atall. But even if we postulated the |ψ|2 distribution at some point in time (e.g., the bigbang), so that the theory would make unambiguous predictions, the distribution of theoutcome Z of an experiment will not be given by a POVM on Hsys, presumably not evenapproximately, and we see no reason why the empirical contents could be summarizedby a formalism at all.

4.2.2 Bohm’s Law and a Modified GRW Law

To improve GRWp1, one may think of modifying it a bit: Instead of choosing the collapsecenter X at random, as prescribed by the GRW process, one could take

X = QI(T ) , (32)

so that the collapse is centered at the actual position of the corresponding particle atthe time of the collapse. In other words, for every collapse the time T and the label Iare chosen at random as in the GRW process, but the position of the (center of the)collapse is not, but is taken from the particle configuration instead.

Let us call this model GRWp2; it was called GRWp in [3]. The behavior of a GRWp2

world is less catastrophic than that of a GRWp1 world, but still the |ψ|2 distributionis not equivariant, and so probabilities cannot be expected to agree with |ψ|2. As withGRWp1, one could consider GRWp2 with |ψ|2 distribution at the big bang, but as withGRWp1, one presumably obtains no POVM, and no formalism.

What this example illustrates: The particle ontology can be combined with a mul-titude of possible laws for the wave function, each of which is simple and respects the

16

symmetries of the GRW process (invariance under rotations, translations, time transla-tions, and Galilean boosts).

With a little modification in its defining equations, GRWp2 becomes a better behavedtheory GRWp3 [10, 48]: Instead of (32), take the collapse center X to be

X = QI(T ) + Z , (33)

where Z is a random 3-vector that is chosen independently of the past with a Gaussiandistribution with mean 0 and covariance matrix diag(σ2, σ2, σ2). It then follows [48]that the conditional distribution of Q(t), given the X, I, T for all collapses up to time t(or given ψt), equals |ψt|2; that the joint distribution of the ψt for all t ≥ 0 is the sameas for the GRW process; and that this theory is empirically equivalent to GRWm andGRWf.

What this example illustrates: The empirical content (i.e., the sum of the empiricalpredictions) of the GRWf and GRWm theories can as well be obtained with the particleontology, not only with the flash and matter density ontologies.

4.2.3 Trajectories From the GRW Wave Function

For the next theory, GRWp4, let us return to the GRW law for ψt, and consider a particleontology with positions given by the wave function by means of the law

Qi(t) = 〈ψt|Qi|ψt〉 , (34)

where Qi is the position operator of particle number i. That is, the actual position Qi(t)is what would in orthodox quantum mechanics be the average position of particle i (ifmeasured). In contrast to GRWp1 and GRWp2 (and Bohmian mechanics), this theorydoes not require any initial data about the particle configuration, as the configurationis a function of ψt. Like GRWp1 and GRWp2, this theory GRWp4 is not empiricallyadequate.

This is best seen when considering a system of identical particles, whose wave func-tion obeys permutation symmetry or anti-symmetry. As a preparation, we note that theGRW process as we have defined it in Section 2.1 above does not preserve permutationsymmetry. For identical particles the GRW process needs to be defined differently toinclude a collapse mechanism which preserves permutation symmetry; for the technicaldetails (which will not matter for our purposes) see [44]. So consider the theory definedby that process together with (34). For a wave function with appropriate permutationsymmetry, (34) implies that all particles (of the same species) have the same position.However, it is empirically incorrect that all electrons have the same position.

What this example illustrates: Apart from being an example of the multitude ofpossible laws for the PO, an interesting trait of this theory is that the configuration ofthe PO supervenes on the wave function by means of the law (34), as it does in GRWm

17

and GRWf. GRWp4 (or the version of it for identical particles) shows that among themany ways in which a configuration of matter (be it a particle configuration, a continuousmatter distribution, or flashes) can supervene on the wave function, different possibilitiesmay strongly disagree about the empirical predictions.

4.2.4 Configuration Jumps and GRW Law

Another theory, GRWp5, has the following laws: The wave function ψt follows the GRWprocess, and the configuration moves according to Bohm’s law of motion (29) betweenthe GRW collapses. However, at the time T when ψ collapses around X ∈ R

3 with labelI, also the configuration Q jumps; more precisely, only the I-th particle jumps, and itjumps to the random center X of the GRW collapse:

QI(T+) = X . (35)

Again, |ψ|2 is not equivariant, and indeed, the behavior of the particles is quitecatastrophic: If ψ(T−) is the wave function of Schrodinger’s cat, the configurationQ(T−) is that of a live cat, and the wave function collapses to that of a dead cat, itmay do so with just a few collapses connected to a few particle labels, corresponding toparticles that would have to be in different positions depending on whether the cat isdead or alive. As a consequence, the configuration after these few collapses will be oneof a live cat, with a few particles moved to where they would have to be if the cat weredead. So, this configuration is very different from what one would normally associatewith |dead〉. Also, this configuration may be well outside the support of both |dead〉 and|alive〉; it may be a configuration for which |ψ|2 is literally zero, or much smaller thaneven the remains of |alive〉 after the collapse. And the behavior of such configurationsshould be expected to be catastrophic.

What this example illustrates: One normally thinks that when ψ = |dead〉 then thereis a dead cat. However, as GRWp5 illustrates, in the PO view this cannot be taken forgranted but must be checked.

4.2.5 Another Way of Configuration Jumps and GRW Law

Within the framework that ψt follows the GRW process and the configuration movesaccording to Bohm’s law of motion (29) between the GRW collapses, further optionsbesides GRWp5 come to mind. Instead of (35), we may postulate that at the time Twhen ψ collapses around X ∈ R

3 with label I, the I-th particle jumps to a randomposition distributed with density

P(QI(T+) ∈ dx)

dx=

∣∣ψT+(Q1(T ) . . .QI−1(T ), x, QI+1(T ) . . . QN(T )

)∣∣2∫dx′

∣∣ψT+(Q1(T ) . . .QI−1(T ), x′, QI+1(T ) . . .QN(T )

)∣∣2 , (36)

which is the |ψT+|2 distribution conditionalized on the configuration of the other parti-cles.

18

Another possibility, which we call GRWp6, is that not just the I-th particle (the oneassociated with the collapse) jumps, but all particles jump. Specifically, choose Q(T+)at random with distribution |ψT+|2.

In GRWp6, the |ψ|2 distribution is indeed equivariant, in the sense that the con-figuration Q(t) will always have distribution |ψt|2. As a consequence, this theory ispresumably empirically equivalent to GRWm and GRWf (in the sense that there is noexperiment that could distinguish GRWp6 from GRWm and GRWf). However, the par-ticles in GRWp6 do not necessarily behave in a reasonable way: For example, consideran agglomerate of N = 1011 particles and arrange a superposition of two well-separatedlocations, ψ = 1√

2

(|here〉⊗N + |there〉⊗N

). Suppose the particles are all “here” (this

happens with probability 1/2) and that the first collapse, which occurs after about 1day, reduces ψ to |there〉⊗N (this happens independently with probability 1/2). In thiscase all particles jump from “here” to “there.”

What this example illustrates: Even if a theory is empirically equivalent to a reason-able theory (such as GRWm and GRWf) it need not itself be a reasonable theory.4

4.3 MBM: Bohm-Like Trajectories From the Master Equation

The most interesting example in our list of toy theories is perhaps MBM; the abbre-viation stands for “master equation Bohmian mechanics.” MBM resembles Bohmianmechanics in that it is deterministic and that its PO consists of particles, but at thesame time it is empirically equivalent to GRWf and GRWm, as we will show in Sec-tion 4.3.1.

The law of motion (29) is replaced by the following equation (considered already in[13, 26]) using, in the role of the wave function, a density matrix ρ:

dQk

dt= vρk(Q1, . . . , QN) =

~

mkIm

∇qk〈q|ρ|q′〉〈q|ρ|q′〉

∣∣∣q=q′=(Q1,...,QN )

. (37)

The density matrix ρ evolves according to (7), which we repeat here for convenience:

dρtdt

= − i~[H, ρt] + λ

N∑

k=1

∫d3x g

1/2k,x ρt g

1/2k,x −Nλρt . (7)

We make a few comments on how these equations are to be understood.Eq. (37) is the natural generalization of Bohmian mechanics to density matrices, and

reduces to (29) in case of a pure state ρ = |ψ〉〈ψ|. However, it is important to noticethat the density matrix considered here is not the one that is normally regarded as thedensity matrix of a system, which arises by averaging |ψ〉〈ψ| (in case the wave functionψ is random) or by tracing out the environment of the system. The density matrix in

4This point is also illustrated by the following theory [14] that is empirically equivalent to Bohmianmechanics: Using a Schrodinger (i.e., non-collapsing) wave function ψt and a particle ontology, let theconfiguration Q(t) be random with distribution |ψ|2, independently of the past.

19

(37), in contrast, does not arise from averaging or partial traces but is assumed to beone of the fundamental variables of the theory. The complete description of the stateis, instead of the pair (Q,ψ) in Bohmian mechanics, the pair (Q, ρ).

The master equation (7) is, in the GRW theories, a consequence of the GRW evo-lution of the wave function. This is different in MBM. In MBM there is no randomwave function. In MBM, (7) holds by fiat, not as a theorem. The defining equations ofMBM—its postulates—are (7) and (37).

What this example illustrates: MBM shows that the empirical content of GRWm andGRWf is compatible with a deterministic theory, and in particular does not imply wavefunction collapse: after all, MBM involves the master equation (7) but not literal wavefunction collapse as in (1). If experiments will confirm the GRW deviations from quan-tum mechanics, then Bohmian mechanics can be modified so as to reproduce the GRWpredictions. MBM also illustrates how it can make sense to speak of the density matrixof the entire universe and, more specifically, how a density matrix can be a fundamentalobject and part of the ontology, rather than just encoding statistical information.

4.3.1 Empirical Equivalence of MBM with GRWm and GRWf

We now prove the empirical equivalence. The derivation of predictions from MBM isanalogous to that from Bohmian mechanics. The analogue in MBM of the quantumequilibrium distribution pψ described in (31) is the distribution

pρ(q) = 〈q|ρ|q〉 . (38)

Note that pρ(q) ≥ 0, and∫Q p

ρ(q) dq = tr ρ = 1. As we show in Appendix A, thisdistribution is equivariant in MBM as a consequence of (37) and (7). It is thereforeconsistent to assume, as we do, that the configuration Qt of the world has distributionpρt at every time t. And therefore, the probability at time t of a certain macroscopicconfiguration is

p(S) =

∫

S

dq 〈q|ρt|q〉 = tr(ρtP (S)

)(39)

where S is the set of all microscopic configurations consistent with that macroscopicconfiguration, and P (S) the projection operator corresponding to the set S, defined by

P (S) =

∫

S

dq |q〉〈q| . (40)

In GRW theories, p(S) can be written in terms of the GRW wave function ψt as

tr(ρt P (S)

)=

∫

H

P(ψt ∈ dφ) ‖P (S)φ‖2 . (41)

Since ψt is typically concentrated on a single macro-configuration, the probability dis-tribution P(ψt ∈ dφ) is typically concentrated on those φ with either ‖P (S)φ‖ ≈ 0 or‖P (S)φ‖ ≈ 1; thus, p(S) equals the probability that ψt is (nearly) concentrated on S.

20

And in this case, either the flashes of GRWf or the matter density of GRWm gives riseto the same macroscopic appearance as configurations from S. In other words, at anyfixed time the MBM, GRWf, and GRWm worlds have the same probability distributionover the possible macro-states.

Now empirical equivalence follows immediately: If there were an experiment whichhad (probably) one outcome Z = z1 in MBM and another one, Z = z2 in GRWm andGRWf, then at the time when the experiment is finished, the probability distributionover the macro-states would have to be different in MBM than in GRWm and GRWf,but it is not.

As a consequence of the empirical equivalence between MBM, GRWm, and GRWf,the empirical content of MBM is summarized by the GRW formalism [34]. (In fact, theGRW formalism was first discovered starting from MBM.) Our argument concerningempirical equivalence also exemplifies that empirical equivalence is a statement aboutthe PO, as discussed in Section 3.

4.4 Master Equation and Matter Density

If one can consider a version of Bohmian mechanics in which the density matrix playsexactly the role of the wave function, then why not do the same trick with the matterdensity and flash ontologies?

For the matter density ontology, this would mean to postulate, in analogy to andreplacing (10),

m(x, t) = tr(ρt M(x)

)(42)

with M(x) =∑

imi δ(Qi − x) the mass density operator, as before. Here, ρt is takento evolve according to the master equation (7). We thus obtain a theory that couldbe called Mm, in which the fundamental objects are a density matrix (which, as inMBM, does not represent an ensemble, or the observer’s limited knowledge, but is, bypostulate, a fundamental object) and the continuous matter with density m(x, t). Thistheory, though, is very different from GRWm! It has a many-worlds character. Forexample, if at some initial time ρ = |ψ〉〈ψ| with |ψ〉 = (|dead〉 + |alive〉)/

√2 being the

wave function of Schrodinger’s cat, then after a short while the GRW function ψt will beeither |dead〉 or |alive〉, but ρt will be |dead〉〈dead|+ |alive〉〈alive|, up to a factor 1

2. As a

consequence, the m field of GRWm will be either mdead or malive, but the m field of Mmwill be mdead +malive, up to a factor 1

2. Both cats are there at once, but with reduced

mass (which the cats, however, do not notice). A very similar theory “Sm”, with theunitary Schrodinger evolution instead of the master equation (7), has been describedin some detail in [3, 4]. It seems possible, perhaps even likely, that Sm is empiricallyequivalent to standard quantum mechanics [4]; if so, then for the same reasons Mmshould be empirically equivalent to GRWm and GRWf.

What this example illustrates: Foremost, this example illustrates the big difference,for quantum theories without observers, between a density matrix and a random wave

21

function. Since with every probability distribution over wave functions there is asso-ciated a density matrix, and since for the purpose of computing predictions only thedensity matrix is relevant, it is common practice in quantum mechanics to immediatelyreplace every probability distribution over wave functions by the density matrix. Here,however, the distinction is crucial: The deterministic m function obtained from thedensity matrix provides a many-worlds picture of reality, whereas the random m func-tion obtained from a random wave function that is either |dead〉 or |alive〉 provides asingle-world picture.

4.5 Master Equation and Flashes

What happens to GRWf when we replace the wave function by a density matrix? Twoversions of what this might mean come to mind:

MGRWf: Postulate that the universe has, at every time t, a density matrix ρt, whoseevolution will be described below. Postulate further that flashes occur, as in GRWf, atrandom times T with constant rate Nλ, and for a random I ∈ 1, . . . , N. In contrastto GRWf, the probability density of the flash location X is given by

p(x) =P(X ∈ dx)

dx= tr

(ρT− gI,x

)(43)

instead of ‖ψT− g1/2I,x ‖2 in GRWf. Postulate further that when a flash occurs at time T ,the density matrix ρT− changes to

ρT+ =1

Cg1/2I,X ρT− g

1/2I,X (44)

with C a normalizing constant. Between collapses, ρt evolves unitarily as usual. Notethat if ρ0 = |ψ0〉〈ψ0| is a pure state then it will remain pure and in fact evolve accordingto the usual GRW dynamics.

It follows that the joint distribution of all flashes is given by the probability measure

PMGRWf(F ∈ S) = tr(ρ0G0(S)

)(45)

with S any set of flash histories and G0(·) the POVM governing the distribution of theflashes in GRWf, see (21). As a consequence of (45), the theory MGRWf, althoughformulated in terms of a density matrix, is physically equivalent to GRWf!

Before justifying this claim, let us elucidate the notion of “physical equivalence,”which can be given a clear definition in the framework of PO [3]: Consider a theoryT (e.g., GRWf with a particular choice of λ, σ, N , and potential V ) and initial dataD0 for T (e.g., for GRWf, an initial wave function ψ0); together, T and D0 define aprobability distribution P over histories of the PO after the initial time (e.g., for GRWf,a distribution over flash patterns in space-time). Now we say that the pair (T ′, D′

0) isphysically equivalent to (T,D0) if it defines the same distribution P as (T,D0). That is,two descriptions of a universe are physically equivalent if they provide the same history

22

of the PO, or (if appropriate) the same probability distribution thereof; variables (suchas ψ) that are not part of the PO may be different in the two descriptions.

For the purpose of comparison with MGRWf, suppose that, as the initial data D0 forGRWf, we do not specify the initial wave function but instead a probability distributionµ over initial wave functions. That still defines a distribution P over the PO histories:Since the joint distribution of all flashes, given ψ0, is 〈ψ0|G0(·)|ψ0〉, the unconditionaljoint distribution of all flashes is

PGRWf(F ∈ S) =

∫

S(H )

µ(dψ) 〈ψ0|G0(S)|ψ0〉 = tr(ρ0G0(S)

), (46)

with ρ0 the density matrix of the ensemble µ. Since this is the same formula as (45),we obtain the physical equivalence. That is, the further possibilities of initial data ρ0 inMGRWf do not lead to new flash histories.

How is MGRWf related to the master equation (7)? Since ρt is random, it does notevolve according to (7). However, there is another density matrix, namely

ρt = Eρt , (47)

where E means the expectation over the random flashes; ρt does evolve deterministicallyaccording to the master equation (7).

What this example illustrates: First, it illustrates the concept of physical equivalence:There is no physical difference between GRWf with a random initial wave function of theuniverse and MGRWf, as the distribution of the PO is the same. Second, it illustratesthe different roles that a density matrix can play: On the one hand, it can be partof the ontology as one of the fundamental objects, such as the random density matrixρt. On the other hand, the other density matrix ρt is a mathematical object encodinginformation about the probability distribution of ρt. Thus, finally, it also illustratesthat it is not necessarily appropriate to speak of “the” density matrix.

Mf: Another way of replacing the wave function in GRWf by a density matrix is toretain (43) for the distribution of the flash location X but not adopt the collapse rule(44). Instead, let ρt evolve deterministically according to the master equation (7). Inother words, Mf arises from MGRWf by replacing ρt in (43) by ρt.

This theory, too, has a many-worlds character, as the set F of flashes will be theunion F = Fdead ∪Falive of a set of flashes of a live cat and a set of flashes of a dead cat(similar to the model “Sf” considered in [4]). For the same reasons as for Mm, Mf ispresumably empirically equivalent to GRWf and GRWm.

What this example illustrates: By way of contrast with MGRWf, it illustrates thedifference between the “collapsing” density matrix of MGRWf and the deterministic onearising from the master equation (7). It thus illustrates why it is important that thelaw governing the PO be precisely formulated (since in particular, it specifies preciselywhich density matrix to use).

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5 Conclusions

In this paper we have illustrated the notion of PO (primitive ontology, i.e., variablesdescribing the distribution of matter in space and time) and its usefulness for a cleanderivation of empirical predictions. We have done so by (i) describing several toy exam-ples of theories with a PO (Section 4), (ii) studying their predictions (Section 4), (iii)reviewing known theories with a PO (Sections 2 and 4.1), and (iv) reviewing knownderivations of predictions (Section 3). In so doing we have illustrated the different rolesthat the PO, the wave function, and the density matrix have in such theories.

A Proof of Equivariance in MBM

We show that equivariance of pρ(q) = 〈q|ρ|q〉 follows from (7) for the Hamiltonian (4)and (37). Before we give the formal proof, we note that the essential reason is that thenon-unitary (diffusive) terms in (7) (i.e., the second and third term on the right handside), do not contribute to the continuity equation for pρ. This can be understood bynoting that a GRW collapse does not change the diagonal entries 〈q|ρt|q〉 of the densitymatrix in the position representation.

Here is the equivariance proof.

∂

∂tpρt(q) = − i

~〈q|[H, ρt]|q〉 −Nλ〈q|ρt|q〉+ λ

N∑

k=1

∫d3x 〈q|g1/2k,x ρt g

1/2k,x |q〉 (48)

=

n∑

k=1

i~2mk

〈q|[∇2qk, ρt]|q〉 −Nλ〈q|ρt|q〉+ λ

N∑

k=1

∫d3x 〈q|ρt|q〉

e−(qk−x)2

2σ2

(2πσ2)3/2(49)

=

n∑

k=1

i~2mk

[(∇2

xk−∇2

yk)〈x|ρt|y〉

]x=y=q

−Nλ〈q|ρt|q〉+Nλ〈q|ρt|q〉 (50)

=

n∑

k=1

i~2mk

[(∇xk +∇yk) · (∇xk −∇yk)〈x|ρt|y〉

]x=y=q

(51)

=n∑

k=1

i~2mk

∇qk ·[(∇xk −∇yk)〈x|ρt|y〉

]x=y=q

(52)

= −n∑

k=1

~

mk∇qk · Im

[∇xk〈x|ρt|y〉

]x=y=q

(53)

= −n∑

k=1

∇qk · (pρt vρtk ) . (54)

Note that the potential V in H does not contribute in (49) because

〈q|V ρt|q〉 =∫dq′ 〈q|V |q′〉︸ ︷︷ ︸

=δ(q−q′)V (q)

〈q′|ρt|q〉 = V (q)〈q|ρt|q〉 = 〈q|ρtV |q〉 . (55)

24

Since any probability distribution p on configuration space will be transported, underthe flow (37), according to the continuity equation

∂p

∂t= −

N∑

k=1

∇qk · (p vρk) , (56)

we have that pt = pρt satisfies (56), which is equivariance.

Acknowledgments. S. Goldstein and R. Tumulka are supported in part by grant no. 37433from the John Templeton Foundation. R. Tumulka is supported in part by NSF GrantSES-0957568 and by the Trustees Research Fellowship Program at Rutgers, the StateUniversity of New Jersey. N. Zanghı is supported in part by INFN.

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